No. Turing equivalence is not enough.
Turing equivalence allows you to compute the same things, true, but it does not allow you to _do_ the same things. For instance if you take brainf*ck, well known to be turing complete, how do you get brainf*ck to do a syscall? You can't.
Besides the computational power you also need the "functional" power. In Unix this means that you need syscalls. If you have syscalls then you have it all. (Since I/O for instance translates into syscalls).
On a more theoretical note you might observe that the Turing machine has an "infinite tape". So any machine that is equivalent to a turing machine will have to have infinite storage capacity.
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